Now, under the conditions of the theorem, we have from Proposition  that the columns of \(\mathbf {M}\)  are asymptotically normal with the same covariance matrix \(\Sigma _{\mathbf {n}^*, f, \mathbf {F}_p}\) . Applying the usual large-sample theory from [1]} to the sequence of mutually conditionally independent statistics \(\overline{\mathbf {S}}_1, \ldots , \overline{\mathbf {S}}_{N}\)  gives the result.
